The Isotropy of Compact Universes

TL;DR

This paper investigates how the topology of the universe influences its isotropy, showing that certain topologies enforce isotropy and analyzing the degrees of freedom in different Bianchi models with compact spaces.

Contribution

It demonstrates that compact topologies eliminate anisotropic modes preventing isotropy and compares the degrees of freedom in Bianchi types I and VII_0 with compact topologies.

Findings

Compactness excludes anisotropic modes in Bianchi VII_h universes.

Type VII_0 solutions are more general than type I in perfect fluid systems.

For specific topologies, the fluid velocity must be non-tilted.

Abstract

We discuss the problem of the stability of the isotropy of the universe in the space of ever-expanding spatially homogeneous universes with a compact spatial topology. The anisotropic modes which prevent isotropy being asymptotically stable in Bianchi-type $VII_h$ universes with non-compact topologies are excluded by topological compactness. Bianchi type $V$ and type $VII_h$ universes with compact topologies must be exactly isotropic. In the flat case we calculate the dynamical degrees of freedom of Bianchi-type $I$ and $VII_0$ universes with compact 3-spaces and show that type $VII_0$ solutions are more general than type $I$ solutions for systems with perfect fluid, although the type $I$ models are more general than type $VII_0$ in the vacuum case. For particular topologies the 4-velocity of any perfect fluid is required to be non-tilted. Various consequences for the problems of the isotropy, homogeneity, and flatness of the universe are discussed.